Summary: Math 1550 Fall 2005
Section 31 P. Achar
Due: December 6, 2005
The dimension of a geometric object is the number of coordinates it takes to specify a location in the object.
For example, the surface of a sphere (such as the surface of the earth) is 2-dimensional, because you can
specify a location with two numbers: its latitude and longitude. On the other hand, a solid sphere is 3-
dimensional: to identify a point within the earth, you must give its latitude, longitude, and depth from the
surface. The way in which one speaks of the "size" of an object depends on its dimension: 1-dimensional
objects have length, 2-dimensional things have area, and 3-dimensional things have volume.
It is possible to study the geometry of objects with more than three dimensions, even though we obviously
can't imagine or draw pictures of such objects. There aren't any ordinary English words to describe things
in more than three dimensions, however, so what mathematicians usually do is just use 3-dimensional
terminology, but with added numerical prefixes to indicate the dimension. For instance, there is no English
word for the "size" of a 4-dimensional object (which would be measured in cm4
), so mathematicians
just call it "4-volume." You can also use "1-volume" as a synonym for "length," and "2-volume" as a
synonym for "area."
Circles and spheres are part of a family of shapes in all different dimensions. The goal of this problem is to